Study on the influencing parameters of rough-strip energy dissipators of curved spillways based on orthogonal tests and numerical simulation

h

Average R-SED height

α

R-SED angle

s

R-SED spacing

w

Bend width

R

Bend centerline radius

Q

Discharge flow rate

i

Slope along the course

L₁

Length of the straight inlet section

L₂

Length of the straight outlet section

h₁

R-SED height at the concave bank

h₂

R-SED height at the convex bank

O

The center of the curvature of the bend

δ

R-SED thickness

H

Water depth at the measurement point

H₀

Head over the weir

C₀

Discharge coefficient of the weir

P₁

Height of the weir

w₁

Width of the flow diversion canal at the upstream weir

η

Energy dissipation rate

η_(i)

Energy dissipation rate under the ith condition

TE₁

Total mechanical energy per unit weight of water in the upstream flow cross-section

TE₂

Total mechanical energy per unit weight of water in the downstream flow cross-section

SS

Water surface superelevation coefficient

SS_(i)

Water surface superelevation coefficient under the ith condition

σ_i

Standard deviation of the superelevation of transverse water surface (Δφ) of all calculated sections under the ith condition

μ_i

Mean value of superelevation of transverse water surface of all calculated sections under the ith condition

Δφ_ij

Difference between the water surface at the concave bank of the bend and the horizontal plane across the center at the jth calculated section under the ith condition

w_i

Water surface width of the open channel based on the centerline water surface elevation under the ith condition

Δφ

Transverse water surface superelevation of the bend

W_j

Entropy weight

p_ij

Proportion of the ith evaluation object under the jth index

E_j

Entropy value of the jth index

⁠,

Euclidean distance and between each object and the ideal solution

C

Relative closeness

P

Significance factor

DF

Degree of freedom

Adj SS

Adjusted sum of squared deviation from mean

Adj MS

Adjusted mean squared error

F

Chi-square test factor

η*

Membership degree of energy dissipation (η)

SS*

Membership degree of water surface superelevation coefficient (SS)

k

Turbulent kinetic energy

ε

Turbulent kinetic energy dissipation rate

α_w

Volume fraction of water in the calculation area with respect to the calculation element

α_α

Volume fraction of air in the calculation area with respect to the calculation element

T

Turbulent kinetic energy

The spillway is a key component of reservoir discharge structures (Kells & Smith 1991; Zhang et al. 2015; Damarnegara et al. 2020), mainly employed for energy dissipation and flood control. Influenced by topographic and geological conditions, construction conditions and engineering economy, some spillways must have corners, thus forming bends at the corner (Seo & Shin 2018; Yang et al. 2019). The water flowing through the bend is called bend flow, which is different from straight-section flow. When water flows through a bend, centrifugal inertia forces are generated by the curved movement of the bend flow. Thus, the surface and bottom flow migrate to the concave and convex banks, respectively, forming a closed transverse circulation within the bend section (Seyedashraf & Akhtari 2015). This circulation is combined with the longitudinal flow, forming a spiral flow. This spiral flow moves forward in the mainstream direction (Johannesson & Parker 1989). In addition, the water depth increases at the concave bank of the bend while decreasing at the convex bank. This causes a transverse gradient of the water surface (Dietrich et al. 1979).

Since Thomson (1876) revealed the existence of both longitudinal and lateral motion in bend flow, studies on bend flow have focused on two aspects, i.e., hydraulic characteristics and engineering measures. The studies on the hydraulic characteristics of bend flow mainly focus on the water depth distribution (Bathurst & Hey 1979; Molls & Chaudhry 1995; Qin et al. 2016; Zhou et al. 2017; Maatooq & Hameed 2020), flow velocity distribution (De Vriend & Geldof 1983; Anwar 1986; Odgaard & Bergs 1988; Ye & McCorquodale 1998; Han et al. 2011; Vaghefi et al. 2015; Moncho-Esteve et al. 2018; Pradhan et al. 2018; Schreiner et al. 2018; Hu et al. 2019; Kim et al. 2020; Yan et al. 2020) and secondary flow evolution (Jin & Steffler 1993; Booij 2003; Huai et al. 2012; Ramamurthy et al. 2013; Engel & Rhoads 2016; Gu et al. 2016; Shaheed et al. 2021). These studies generally describe the characteristics and evolution of bend flow and provide a theoretical basis for further studies. Based on the hydraulic characteristics of bend flow, some engineering measures in the bend have been proposed using physical model tests or numerical simulation in order to improve the bend flow structure. For example, Zhang et al. (2015) arranged continuous guide walls at the central axis of the bend to improve the bend flow pattern. However, the height of the guide wall needs to be determined based on the water depth at the bend inlet. The guide wall only applies to the bend with a small centerline radius. Thus, the practicality needs to be improved. Yang et al. (2019) arranged permeable spur dikes at the concave bank of the bend to improve the flow pattern of ‘backwater at the concave bank’. The locations with permeable spur dikes are significantly prone to congestion, which is not conducive to the structural stability of permeable spur dikes during long-term operation. Ranjan et al. (2006) arranged vanes at the central axis of the bend to reduce the secondary flow intensity of the bend flow. However, the vanes do not have energy dissipation effects and cannot dissipate the excess energy of the bend flow. Martin-Vide et al. (2010) arranged ripraps at the concave bank of the bend to reduce the scouring of bend flow at the concave bank. However, this did not change the flow pattern of ‘increasing water depth at the concave bank and decreasing water depth at the convex bank’ in the bend. In this paper, the rough-strip energy dissipators (R-SEDs) were arranged at the bend bottom of the curved spillway. These R-SEDs are simple in shape and convenient in construction and have achieved better performance in improving the flow structure and energy dissipation in the bend (Zhang et al. 2022a, 2022b).

The study on R-SEDs was mainly based on the hydraulic model test of the curved spillway of the Yin'ejike 635 Reservoir in Xinjiang, China (the geometric scale was 1:50). The physical model test was completed in the curved spillway flume of the Xinjiang Key Laboratory of Water Conservancy Engineering Safety and Water Disaster Prevention, China. At a discharge flow rate of 800 m³/s, the water depth and flow velocity at the concave and convex banks were significantly different and the bend flow was turbulent. To tackle these adverse flow patterns, R-SEDs have been arranged at the bend bottom and have shown high effectiveness in energy dissipation and flow diversion (Zhang et al. 2022b). Since then, the R-SEDs have been increasingly investigated. For example, Zhang et al. (2022a) mainly analyzed the results of the R-SEDs in the hydraulic model test of the curved spillway of the Yin'ejike 635 Reservoir using the single-factor test method. They only considered the effects of R-SED arrangement parameters on the energy dissipation and flow diversion effects of R-SEDs. The effects of engineering parameters of the curved spillway on these effects were not included. Zhang et al. (2022b) derived an equation for calculating the energy dissipation rate of R-SEDs through physical model tests, without considering the flow diversion effect of R-SEDs on bend flow. Thus, these studies still have some limitations.

Therefore, in this paper, the R-SEDs were arranged at the bend bottom of the curved spillway. Orthogonal tests and numerical simulation were performed to evaluate the R-SED arrangement parameters (average R-SED height h, R-SED angle α, R-SED spacing s) and the spillway engineering parameters (bend width w, bend centerline radius R, discharge flow rate Q) that affect the energy dissipation and diversion effects of R-SEDs. The energy dissipation rate, water surface superelevation coefficient and relative closeness were selected as response variables. Based on the entropy weight and technique for order performance by similarity to ideal solution (TOPSIS) methods, a multi-criteria evaluation system was established to analyze these parameters and determine their optimal values. Thus, an optimal parameter combination was recommended, i.e., h = 1.8 cm, s = 24 cm, α = 18°, R = 200 cm, w = 80 cm and Q = 20.0 L/s. This can provide a reference for the R-SED design of similar curved spillways. In addition, the established water surface superelevation coefficient calculation model is simple and easy to understand, facilitating its application in engineering design.

Orthogonal experimental design, referred to as orthogonal design, is a method to scientifically arrange and analyze multi-factor tests using orthogonal tables. It is one of the most commonly used experimental design methods. In the orthogonal design, representative points that meet the requirements of balance and orthogonality can be selected from the whole test and tested so as to obtain the overall stability of the development trend (Wang et al. 2022; Yuan et al. 2022).

Considering the influence of R-SED arrangement parameters and spillway engineering parameters on the energy dissipation and flow diversion effect of R-SEDs, six influencing factors were selected, including average R-SED height (h), R-SED spacing (s), R-SED angle (α), bend centerline radius (R), bend width (w) and discharge flow rate (Q). Based on the previous findings of installing R-SEDs (Zhang et al. 2022a, 2022b), three levels were selected for each factor: (−1), 0 and (+1). The orthogonal table L₁₈(3⁷) was selected for the test design. The factors and levels of the orthogonal test are shown in Table 1. In addition, the orthogonal test does not consider the effect of interlevel interactions. Thus, the evaluation index is only affected by the factors' independent effects (i.e., main effects) and random errors (i.e., residuals).

Test measurement instruments were selected and arranged according to the Chinese standard ‘Calibration Method of Common Instruments for Hydraulic and River Model Test’ (SL 233-2016) (Ministry of Water Resources of China 2016).

• (a)

  Water depth measurement

The water level measurement probe was used to measure the water depth with an accuracy of 0.1 mm. A total of 51 water depth measurement cross-sections were arranged along the spillway model, i.e., #0 ∼ #50. Each cross-section was arranged with 11 measurement points (i.e., A ∼ K). To reduce the disturbance of the sidewall to the flow, two near-bank measurement points (A and E) were located 1 cm from the sidewalls of the concave and convex banks, respectively.

• (b)

  Flow velocity measurement

The flow velocity was measured using a Pitot tube, with an accuracy of 0.1 mm. A total of 13 measurement cross-sections (i.e., #0, #4, #8, #12, #16, #20, #24, #28, #32, #36, #40, #44 and #48) were selected as flow velocity measurement cross-sections. Six measurement points (i.e., A, C, E, G, I and K) were selected for each cross-section. The vertical measurement position of each measurement point was located at 2H/3 from the bottom (H is the water depth at the measurement point).

The R-SEDs have a dual effect of energy dissipation and flow diversion on the bend flow. Evaluation indexes need to be constructed to assess the energy dissipation effect and flow diversion effect of the R-SEDs in the 18 orthogonal tests.

Cross-sections #8 and #32 were selected as the upstream and downstream sections of the bend, respectively. The horizontal plane where the bottom elevation of Cross-section #32 was located was taken as the reference plane. The energy dissipation rate in the 18 test scenarios was calculated using Equations (3)–(5). The calculation results of the energy dissipation rate are shown in Table 3.

To fully measure the flow diversion effect of R-SEDs, Cross-sections #8 ∼ #48 (41 cross-sections in total) were selected to calculate the water surface superelevation coefficient. The water surface superelevation coefficients were calculated using Equations (6)–(9) for 18 orthogonal tests. The calculation results are shown in Table 3.

• (a)

  Establishing an evaluation system of the combined effect of energy dissipation and flow diversion

Eighteen groups of orthogonal tests were used as feasibility study schemes, and the energy dissipation rate and the water surface superelevation coefficient were used as target variables to construct the original matrix ⁠.

• (b)

  Determining the weight of the energy dissipation rate and the water surface superelevation coefficient using the entropy weight method

In summary, the established multi-criteria evaluation system consists of two sub-evaluation indexes (the energy dissipation rate and water surface superelevation coefficient) and one comprehensive evaluation index (relative closeness).

Firstly, a three-dimensional (3D) spillway model was established using SOLIDWORKS 2018. The 3D model included the straight inlet section, the 60° bend and the straight outlet section, as shown in Figure 1(b). Then, the calculation area of the spillway was extracted and meshed using ANSYS ICEM CFD 19.0. Considering the complex bend structure and free water surface in the spillway, the hybrid structured meshes (tetrahedral and hexahedral elements) were used to mesh the 3D calculation area. The structured meshes have the advantages of arbitrary structure and strong adaptability. Finally, the spillway model was calculated using ANSYS FLUENT 19.0.

The renormalization group (RNG) k-ɛ turbulence model was selected as the turbulence model. Derived from a rigorous statistical technique, this model considers turbulent vortices and can accelerate the calculation of additional terms of the turbulent kinetic energy dissipation rate equation. It has successfully simulated many complex flow problems (Huai et al. 2012; Ramamurthy et al. 2013; Ghazanfari-Hashemi et al. 2019). The two equations of the RNG k-ε turbulence model are as follows:

In this study, numerical simulations of the spillway flow were performed using the RNG k-ε turbulent flow model and the VOF method. The control volume finite element method (CVFEM) method was used to discretize the control equations. The semi-implicit SIMPLE algorithm was used to solve the velocity-pressure field coupling. PRESTO! was selected to calculate the pressure equations. The standard wall function method was used to deal with the near-wall flow.

The boundary condition setting of the spillway model is shown in Figure 7. According to the model test results, the water-air interface was set at the inlet water depth of 0.056 m. Thus, the water and air inlets were delineated. The boundary conditions of the water and air inlets were set as the velocity inlet and the pressure inlet (1 atm). The outlet boundary condition was set as the pressure outlet (1 atm). The standard no-slip solid wall boundary condition was applied to the remaining solid boundaries. The initial inlet water depth and flow velocity were 0.056 m and 0.448 m/s, respectively.

Minitab 21.1 software was used to perform the analysis of variance (ANOVA) and intuitive analysis on the calculated energy dissipation rate, water surface superelevation coefficient and relative closeness for the 18 orthogonal tests.

Using the energy dissipation rate as the response variable, ANOVA was conducted on the six influencing factors (predictor variables). The results are shown in Table 5. The significance level was selected as 0.05. When the P-value corresponding to the main effect was less than 0.05 (P < 0.05), the original hypothesis was rejected, and the total effect of the regression was considered significant. The analysis was focused on the main effects on the evaluation indexes without considering the interactions between different factors. Adjusted R² (R² (adj)) = 83.98% > 80.0% (Table 5) for the influencing factors, indicating that the orthogonal tests were statistically significant. Further analysis of the test levels shows that the differences between the different groups of bend centerline radius (R) and bend width (w) were significant (P_R < 0.05, P_w < 0.05). This indicates that these two factors dominated the changing trend of the energy dissipation rate. The remaining factors had a test level of P > 0.05, indicating that these factors showed homogeneity of variance and did not exhibit significant differences. This indicates that the remaining factors had insignificant effects on the energy dissipation rate. The above analysis shows that the spillway engineering parameters influenced the energy dissipation rate more than the R-SED arrangement parameters. Therefore, the influence of the spillway body shape on the energy dissipation rate of R-SEDs should be highlighted in engineering design.

The characteristic mean of each factor level is first obtained. Then, the main effect plot can be achieved by connecting the response mean of each factor level using a line. Whether a factor has the main effect can be determined by comparing the slope of the regression lines. A larger slope of the regression line indicates more significant main effects. The main effect plot of the energy dissipation rate (η) is shown in Figure 8. The regression lines of bend width (w) and bend centerline radius (R) were the steepest, indicating that η was the most sensitive to the change of these two factors. The two factors (w and R) were positive indicators, i.e., at a larger factor, η is higher, and R-SEDs' energy dissipation effects are larger. η was the highest (0.427) at the largest bend width (w = 80 cm) while decreasing by 31.38% with w decreasing to 50 cm. η was the highest (0.405) at the largest bend centerline radius (R = 200 cm) while decreasing by 23.85% with R decreasing to 140 cm. The slopes of the regression lines of three factors (average R-SED height h, R-SED angle α and discharge flow rate Q) were similar, i.e., the effect of these three factors on η was consistent. These three factors were negative indicators, i.e., at a larger factor, η is lower, and R-SEDs' energy dissipation effects are lower. η was the highest (0.385) at the smallest average R-SED height (h = 1.2 cm) while decreasing by 7.54% with h increasing to 1.8 cm. η was the highest (0.388) at the smallest R-SED angle (α = 18°) while decreasing by 8.99% with α increasing to 26°. η was the highest (0.385) at the smallest discharge flow rate (Q = 20 L/s) while decreasing by 8.15% with Q increasing to 25.0 L/s. R-SED spacing (s) was a two-way indicator, i.e., s is a positive indicator within 18–24 cm and a negative indicator within 24–30 cm. η was the highest (0.389) at s = 24 cm. η increased by 8.06% with s increasing from 18 to 24 cm and increased by 6.28% with s decreasing from 30 to 24 cm.

The water surface superelevation coefficient was used as the response variable. The significance level was selected as 0.05. The ANOVA was performed on the six influencing factors. The results are shown in Table 7. R² (adj) = 92.08% > 80.0% for these influencing factors (Table 7), indicating that the orthogonal test was statistically significant. Further analysis shows that the test levels of average R-SED height (h), R-SED spacing (s) and bend centerline radius (R) differed significantly between groups (P_h < 0.05, P_s < 0.05 and P_R < 0.05). This indicates that these three factors dominated the changing trend of the water surface superelevation coefficient. The remaining factors had a test level of P > 0.05. Thus, these factors had homogeneity of variance and did not show significant differences. Thus, they had fewer effects on the water surface superelevation coefficient. The above analysis shows that compared with the spillway engineering parameters, the R-SED arrangement parameters significantly affected the water surface superelevation coefficient.

The main effect plot of water surface superelevation coefficient (SS) is shown in Figure 9. The regression lines of average R-SED height (h), R-SED spacing (s) and bend centerline radius (R) were the steepest. This indicates that SS was the most sensitive to the changes in these three factors. Average R-SED height (h) was a negative indicator, i.e., at a larger h, SS is higher, and R-SEDs' flow diversion effects are lower. SS was the smallest (0.345) at the smallest average R-SED height (h = 1.2 cm) while increasing by 29.88% with h increasing to 1.8 cm. R-SED spacing (s), bend centerline radius (R) and R-SED angle (α) were positive indicators, i.e., at a larger value, SS is smaller, and R-SEDs' flow diversion effects are larger. SS was the smallest (0.388) at the largest R-SED spacing (s = 30 cm) while increasing by 15.47% with s decreasing to 18 cm. SS was the smallest (0.378) at the largest bend centerline radius (R = 200 cm) while increasing by 19.40% with R decreasing to 140 cm. SS was the smallest (0.406) at the largest R-SED angle (α = 26°) while increasing by 8.56% with α decreasing to 18°. Bend width (w) and discharge flow rate (Q) were two-way indicators. The bend width (w) was a negative indicator within 50–60 cm and a positive indicator within 60–80 cm. SS is the largest at w = 60 cm, with the weakest flow diversion effect. SS decreased by 7.27% with w decreasing from 60 to 50 cm while decreasing by 5.68% from 60 to 80 cm, with improved flow diversion effects. The discharge flow rate (Q) was a positive indicator within 20.0–22.5 L/s and a negative indicator within 22.5–25.0 L/s. At Q= 22.5 L/s, SS was the smallest (0.409), and the R-SEDs had the most significant flow diversion effect.

The weights of each evaluation index (η and SS) were calculated using Equations (10)–(13) and the calculated parameters are shown in Table 9. The weight of the energy dissipation rate (W_(η) = 0.629) was larger than that of the superelevation coefficient (W_(SS) = 0.371). This indicates that the energy dissipation rate carried more information than the water surface superelevation coefficient and that the energy dissipation effect of R-SEDs on the bend flow was larger than the flow diversion effect.

The relative closeness can comprehensively measure the energy dissipation and flow diversion effects of R-SEDs. The relative closeness of each run to the ideal solution was calculated using Equations (14)–(18). Then, the overall ranking of the energy dissipation and flow diversion effect of R-SEDs on the bend flow at each run was obtained (Table 10). Run 5 (h = 1.5 cm, s = 24 cm, α = 22°, R = 200 cm, w = 80 cm and Q = 20L/s) had the largest relative closeness and the highest overall rating.

Table 10

Overall ranking of each run using the TOPSIS method

Run .	η .	SS .	1 − SS .	b(η) .	b(1−SS) .	z(η) .	z(1−SS) .	.	.	C .	Rank .
1	0.305	0.466	0.534	0.191	0.214	0.120	0.080	0.083	0.032	0.279	16
2	0.407	0.359	0.641	0.255	0.198	0.160	0.074	0.054	0.057	0.514	7
3	0.415	0.241	0.759	0.260	0.133	0.164	0.049	0.072	0.055	0.432	11
4	0.358	0.488	0.512	0.224	0.270	0.141	0.100	0.057	0.060	0.513	8
5	0.498	0.386	0.614	0.312	0.213	0.196	0.079	0.035	0.092	0.728	1
6	0.278	0.414	0.586	0.174	0.229	0.110	0.085	0.091	0.035	0.279	16
7	0.354	0.555	0.445	0.222	0.307	0.140	0.114	0.057	0.071	0.557	6
8	0.322	0.456	0.544	0.202	0.252	0.127	0.093	0.072	0.048	0.397	12
9	0.429	0.489	0.511	0.269	0.270	0.169	0.100	0.030	0.079	0.722	2
10	0.396	0.328	0.672	0.248	0.181	0.156	0.067	0.061	0.051	0.451	10
11	0.434	0.408	0.592	0.272	0.225	0.171	0.084	0.039	0.071	0.644	4
12	0.346	0.272	0.728	0.217	0.150	0.136	0.056	0.083	0.028	0.254	18
13	0.332	0.384	0.616	0.208	0.212	0.131	0.079	0.074	0.037	0.331	15
14	0.314	0.467	0.533	0.197	0.258	0.124	0.096	0.075	0.049	0.394	13
15	0.445	0.367	0.633	0.279	0.203	0.175	0.075	0.044	0.071	0.620	5
16	0.413	0.523	0.477	0.259	0.289	0.163	0.107	0.034	0.079	0.698	3
17	0.36	0.438	0.562	0.226	0.242	0.142	0.090	0.059	0.052	0.468	9
18	0.276	0.497	0.503	0.173	0.274	0.109	0.102	0.088	0.052	0.372	14

Run .	η .	SS .	1 − SS .	b(η) .	b(1−SS) .	z(η) .	z(1−SS) .	.	.	C .	Rank .
1	0.305	0.466	0.534	0.191	0.214	0.120	0.080	0.083	0.032	0.279	16
2	0.407	0.359	0.641	0.255	0.198	0.160	0.074	0.054	0.057	0.514	7
3	0.415	0.241	0.759	0.260	0.133	0.164	0.049	0.072	0.055	0.432	11
4	0.358	0.488	0.512	0.224	0.270	0.141	0.100	0.057	0.060	0.513	8
5	0.498	0.386	0.614	0.312	0.213	0.196	0.079	0.035	0.092	0.728	1
6	0.278	0.414	0.586	0.174	0.229	0.110	0.085	0.091	0.035	0.279	16
7	0.354	0.555	0.445	0.222	0.307	0.140	0.114	0.057	0.071	0.557	6
8	0.322	0.456	0.544	0.202	0.252	0.127	0.093	0.072	0.048	0.397	12
9	0.429	0.489	0.511	0.269	0.270	0.169	0.100	0.030	0.079	0.722	2
10	0.396	0.328	0.672	0.248	0.181	0.156	0.067	0.061	0.051	0.451	10
11	0.434	0.408	0.592	0.272	0.225	0.171	0.084	0.039	0.071	0.644	4
12	0.346	0.272	0.728	0.217	0.150	0.136	0.056	0.083	0.028	0.254	18
13	0.332	0.384	0.616	0.208	0.212	0.131	0.079	0.074	0.037	0.331	15
14	0.314	0.467	0.533	0.197	0.258	0.124	0.096	0.075	0.049	0.394	13
15	0.445	0.367	0.633	0.279	0.203	0.175	0.075	0.044	0.071	0.620	5
16	0.413	0.523	0.477	0.259	0.289	0.163	0.107	0.034	0.079	0.698	3
17	0.36	0.438	0.562	0.226	0.242	0.142	0.090	0.059	0.052	0.468	9
18	0.276	0.497	0.503	0.173	0.274	0.109	0.102	0.088	0.052	0.372	14

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Relative closeness was used as the response variable. A significance level of 0.05 was selected. The ANOVA was conducted to examine the importance of the six influencing factors on the relative closeness, as shown in Table 11. R² (adj) = 87.92% > 80.0% (Table 11) for the influencing factors, indicating that the orthogonal test was statistically significant. Further analysis shows that the differences in the test levels between groups were significant for bend width (w), average R-SED height (h) and bend centerline radius (R) (P_w < 0.05, P_h < 0.05 and P_R < 0.05). This indicates that these three factors dominated the changing trend of relative closeness. The remaining factors had a test level of P > 0.05. Thus, these factors showed homogeneity of variance and did not exhibit significant differences. Thus, the remaining factors had insignificant effects on the relative closeness.

The main effect plots of relative closeness (C) are shown in Figure 10. The regression lines of bend width (w), average R-SED height (h) and bend centerline radius (R) were the steepest, indicating that C was the most sensitive to the changes of these three factors. These three factors were positive indicators, i.e., at a larger factor, C is larger, and the comprehensive energy dissipation and flow diversion effects of R-SEDs are stronger. C was the largest (0.613) at the largest bend width (w = 80 cm) while decreasing by 81.90% with w decreasing to 50 cm. C was the largest (0.537) at the largest average R-SED height (h = 1.8 cm) while decreasing by 24.88% with h decreasing to 1.2 cm. C was the largest (0.524) at the largest bend centerline radius (R = 200 cm) while decreasing by 24.76% with R decreasing to 140 cm. R-SED angle (α) and discharge flow rate (Q) were negative indicators, i.e., at a larger factor, C is smaller, and the comprehensive energy dissipation and diversion effects of R-SEDs are weaker. C reached the maximum (0.542, 0.513) when these two factors were 18° and 20.0 L/s, respectively. R-SED spacing (s) was a two-way indicator: a positive indicator within 18–24 cm and a negative indicator within 24–30 cm. At s = 24 cm, C reached the maximum (0.525). Combined with the findings of Section 3.1.2, the two-way nature of R-SED spacing (s) indicates that the value of s should not be too large or too small. Excessively large R-SED spacing will reduce the number of R-SEDs, while too small spacing will reduce the contact area of water flow between adjacent R-SEDs. These are not conducive to energy dissipation and flow diversion.

The value of each factor at which the relative closeness reached its maximum was selected to obtain the combination of recommended parameters, i.e., h = 1.8 cm, s = 24 cm, α = 18°, R = 200 cm, w = 80 cm and Q = 20.0L/s.

To verify the accuracy of the multi-criteria evaluation system (i.e., to verify the advantages of the recommended parameter combination) and the necessity of arranging R-SEDs in the bend, numerical simulation was performed in three runs, including (1) Run without R-SEDs (R = 200 cm, w = 80 cm and Q = 20.0 L/s); (2) Run 5 (the optimal run in the 18 orthogonal tests: h = 1.5 cm, s = 24 cm, α = 22°, R = 200 cm, w = 80 cm and Q = 20.0 L/s); (3) the run with the recommended parameter combination (Run OPC) (h = 1.8 cm, s = 24 cm, α = 18°, R = 200 cm, w = 80 cm and Q = 20.0 L/s).

The above numerical results show that R-SEDs' arrangement in the spillway bend can facilitate good energy dissipation and flow diversion. All the hydraulic indexes in Run OPC were superior to those in Run 5. These verified the reliability of the established multi-criteria evaluation system (i.e., the advantages of the recommended parameter combination) and the necessity of arranging R-SEDs in the bend.

This study focused on the R-SEDs arranged at the bend bottom of the curved spillway. A multi-criteria evaluation system was established for the comprehensive energy dissipation and flow diversion effects of R-SEDs. Based on hydrodynamic theory, orthogonal tests and numerical simulation, the six factors (average R-SED height h, R-SED spacing s, R-SED angle α, bend width w, bend centerline radius R and discharge flow rate Q) affecting the energy dissipation and diversion effects of the R-SEDs were analyzed to obtain an optimal parameter combination. The main conclusions are drawn as follows:

• (i)

  Bend width (w) and bend centerline radius (R) are the significant factors affecting the energy dissipation effect of R-SEDs. These two factors are positively correlated with the energy dissipation rate. The factors are ranked in descending order in terms of their effects on R-SEDs' energy dissipation performance: bend width (w) > bend centerline radius (R) > R-SED angle (α) > R-SED spacing (s) > discharge flow rate (Q) > average R-SED height (h).

• (ii)

  Average R-SED height (h), R-SED spacing (s) and bend centerline radius (R) are the significant factors affecting the flow diversion effect of R-SEDs. Average R-SED height (h) is negatively correlated with the water surface superelevation coefficient, while R-SED spacing (s) and bend centerline radius (R) are positively correlated. The factors are ranked in descending order in terms of their effects on R-SEDs' flow diversion performance: average R-SED height (h) > bend centerline radius (R) > R-SED spacing (s) > R-SED angle (α) > bend width (w) > discharge flow rate (Q).

• (iii)

  Bend width (w), average R-SED height (h) and bend centerline radius (R) are the significant factors influencing the comprehensive energy dissipation and flow diversion effects. These three factors are positively correlated with relative closeness. The factors are ranked in descending order in terms of their effects on R-SEDs' comprehensive performance: bend width (w) > average R-SED height (h) > bend centerline radius (R) > R-SED angle (α) > R-SED spacing (s) > discharge flow rate (Q).

• (iv)

  R-SEDs are simple in shape and convenient in construction. They can effectively improve the various undesirable hydraulic phenomena in the bend flow. The influence weights of the energy dissipation rate (0.629) and the water surface superelevation coefficient (0.371) indicate that the energy dissipation effect of the R-SEDs on the bend flow is more significant than the flow diversion effect.

• (v)

  Among the 18 groups of orthogonal tests, Run 5 shows the highest overall rating. The run with the recommended parameter combination (h = 1.8 cm, s = 24 cm, α = 18°, R = 200 cm, w = 80 cm and Q = 20.0 L/s) from the multi-criteria evaluation system exhibits better hydraulic evaluation indexes than Run 5. For example, the energy dissipation rate (0.591) is 18.67% higher than that in Run 5. The water surface superelevation coefficient (0.306) is 26.14% lower than that in Run 5. The recommended parameter combination can provide a reference for the R-SED design of similar curved spillways based on actual project topography and geological conditions.

• (vi)

  This study mainly considers the effects of the six key factors (average R-SED height h, R-SED angle α, R-SED spacing s, bend width w, bend centerline radius R and discharge flow rate Q) on the energy dissipation and flow diversion effects of R-SEDs. Further studies on other factors (such as bend angle and bottom slope) will be conducted subsequently. Regarding the R-SED design of the same type of curved spillways, the effects of relevant parameters on R-SEDs' energy dissipation and flow diversion effects in this study can provide a reference for selecting the parameter values. For example, based on the recommended parameter values, the main effect analysis results and engineering characteristics, relevant simulation tests can be performed to select the values of significant factors. For the secondary factors, combined with the recommended parameter values and the main effect analysis results, their values can be directly taken based on engineering experience or economic practicality. The relevant parameters used in this study were the dimensions of the physical laboratory model. Thus, these parameters should be scaled up according to the geometric scale (1:50) for actual engineering design.

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant numbers 52269019 and 51769037) and the University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant number XJEDU2017T004). We would also like to thank XSG Editing for their professional editing service.

This work was supported by the National Natural Science Foundation of China (Grant numbers 52269019 and 51769037) and University Research Program Innovation Team Project of Xinjiang Uygur Autonomous Region (Grant number XJEDU2017T004). The first author has received research support from Xinjiang Agricultural University.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.